How to differentiate y=(2+4x^2)^3?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We can solve y=(2+4x^2)^3 in two ways. One is use the chain rule

y=(2+4x^2)^3

y' = 3*8x*(2 + 4x^2)^2

=> y' = 24x*( 4 + 16x^4 + 16x^2)

=> y' = 96x + 384x^5 + 384x^3

Else expand (2+4x^2)^3

y=(2+4x^2)^3

=> 8 + 64x^6 + 96x^4 + 48x^2

y' = 384x^5 + 384x^3 + 96x

The derivative of y=(2+4x^2)^3 is 384x^5 + 384x^3 + 96x

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Since the given function is composed, we'll apply chain rule to differentiate it.

We'll differentiate with respect to x. First, we'll identify the composed functions, whose final result is y.

dy/dx = (dy/dt)*(dt/dx)

We'll put 2+4x^2 = t

y = t^3

We'll differentiate with respect to t:

dy/dt = d(t^3)/dt

dy/dt = 3t^2

We'll differentiate t with respect to x.

dt/dx = d(2+4x^2)/dx

dt/dx = 8x

dy/dx = 3t^2*8x = 24x*t^2

We'll substitute back t:

dy/dx = 24x(2 + 4x^2)^2

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